Question

question 9 (3 points)
given polynomial f(x) and a factor of f(x), factor f(x) completely.
f(x) = x³ + 18x² + 95x + 150 ; x + 10
f(x) = (____)(__)(____)
blank 1:
blank 2:
blank 3:
question 10 (3 points)
given polynomial f(x) and a factor of f(x), factor f(x) completely.
f(x) = x³ - 2x² - 40x - 64 ; x - 8
f(x) = (____)(__)(____)
blank 1:
blank 2:
blank 3:

Explanation:

Question 9

Step 1: Use Polynomial Long Division

We divide \( f(x) = x^3 + 18x^2 + 95x + 150 \) by \( x + 10 \).
Using long division:
\[

$$\begin{array}{r|rrrr} x + 10 & x^3 & +18x^2 & +95x & +150 \\ \hline & x^2 & +8x & +15 & \\ & x^3 & +10x^2 & & \\ \hline & & 8x^2 & +95x & \\ & & 8x^2 & +80x & \\ \hline & & & 15x & +150 \\ & & & 15x & +150 \\ \hline & & & & 0 \\ \end{array}$$

\]
So, \( f(x) = (x + 10)(x^2 + 8x + 15) \).

Step 2: Factor the Quadratic

Factor \( x^2 + 8x + 15 \). We need two numbers that multiply to 15 and add to 8. Those numbers are 3 and 5.
So, \( x^2 + 8x + 15 = (x + 3)(x + 5) \).

Step 3: Write the Completely Factored Form

Putting it all together, \( f(x) = (x + 10)(x + 3)(x + 5) \).

Step 1: Use Polynomial Long Division

We divide \( f(x) = x^3 - 2x^2 - 40x - 64 \) by \( x - 8 \).
Using long division:
\[

$$\begin{array}{r|rrrr} x - 8 & x^3 & -2x^2 & -40x & -64 \\ \hline & x^2 & +6x & +8 & \\ & x^3 & -8x^2 & & \\ \hline & & 6x^2 & -40x & \\ & & 6x^2 & -48x & \\ \hline & & & 8x & -64 \\ & & & 8x & -64 \\ \hline & & & & 0 \\ \end{array}$$

\]
So, \( f(x) = (x - 8)(x^2 + 6x + 8) \).

Step 2: Factor the Quadratic

Factor \( x^2 + 6x + 8 \). We need two numbers that multiply to 8 and add to 6. Those numbers are 2 and 4.
So, \( x^2 + 6x + 8 = (x + 2)(x + 4) \).

Step 3: Write the Completely Factored Form

Putting it all together, \( f(x) = (x - 8)(x + 2)(x + 4) \).

Answer:

Blank 1: \( x + 10 \)
Blank 2: \( x + 3 \)
Blank 3: \( x + 5 \)

Question 10